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Cauchy problem and initial trace for a doubly degenerate parabolic equation with strongly nonlinear sources. (English) Zbl 0993.35057
Let ${S}_{T}={ℝ}^{N}×\left(0,T\right)$, $N\ge 1$ and $T>0$. The author investigates in ${S}_{T}$ the Cauchy problem for the equation ${u}_{t}=div\left(|D{u}^{m}{|}^{p-2}D{u}^{m}\right)+{u}^{q}$ with the initial condition $u\left(x,0\right)={u}_{0}\left(x\right)$. Here $p>1$, $m>0$, $m\left(p-1\right)>1$, $q>1$, and ${u}_{0}$ is locally integrable in ${ℝ}^{N}$. Of course, for $m=1$ we have the familiar evolution $p$-Laplacian equation, and for $p=2$ we have the porous media equation. The Cauchy problem for the general case is investigated for a large class of initial conditions. To describe this class, the following norm is defined for $h\ge 1$. ${|||f|||}_{h}={sup}_{x\in {ℝ}^{N}}{\parallel f\parallel }_{h}\left({B}_{1}\left(x\right)\right)$. Here, ${\parallel ·\parallel }_{h}$ represents the usual norm in ${L}^{h}\left({B}_{1}\left(x\right)\right)$, where ${B}_{1}\left(x\right)$ denotes the unit ball centered at $x$ in ${ℝ}^{N}$. The first result is the following. Assume ${u}_{0}\ge 0$, $|||{u}_{0}{|||}_{h}<\infty$, where $h=1$ if $q and $h>\left(N/p\right)\left(q-m\left(p-1\right)\right)$ otherwise. Then there is a constant ${T}_{0}>0$ depending on the data such that a solution $u\left(x,t\right)$ exists in ${S}_{{T}_{0}}$. Quantitative bounds for the solution and results involving a supersolution are obtained. Also the problem of uniqueness is discussed.
MSC:
 35K65 Parabolic equations of degenerate type 35K15 Second order parabolic equations, initial value problems 35K55 Nonlinear parabolic equations 35B45 A priori estimates for solutions of PDE